{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T19:52:50Z","timestamp":1775245970122,"version":"3.50.1"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T00:00:00Z","timestamp":1571184000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T00:00:00Z","timestamp":1571184000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,12]]},"DOI":"10.1007\/s10915-019-01065-3","type":"journal-article","created":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T13:29:07Z","timestamp":1571232547000},"page":"1945-1962","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Explicit Time Stepping of PDEs with Local Refinement in Space-Time"],"prefix":"10.1007","volume":"81","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7464-455X","authenticated-orcid":false,"given":"Dylan","family":"Abrahamsen","sequence":"first","affiliation":[]},{"given":"Bengt","family":"Fornberg","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,10,16]]},"reference":[{"issue":"25\u201328","key":"1065_CR1","doi-asserted-by":"publisher","first-page":"3247","DOI":"10.1016\/j.cma.2005.06.013","volume":"195","author":"R Abedi","year":"2006","unstructured":"Abedi, R., Petracovici, B., Haber, R.: A space-time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput. Methods Appl. Mech. Eng. 195(25\u201328), 3247\u20133273 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"5","key":"1065_CR2","doi-asserted-by":"publisher","first-page":"A2020","DOI":"10.1137\/16M1084407","volume":"39","author":"M Almquist","year":"2017","unstructured":"Almquist, M., Mehlin, M.: Multilevel local time-stepping methods of Runge\u2013Kutta-type for wave equations. SIAM J. Sci. Comput. 39(5), A2020\u2013A2048 (2017)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1065_CR3","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1007\/s10915-019-01028-8","volume":"81","author":"V Bayona","year":"2019","unstructured":"Bayona, V.: Comparison of moving least squares and RBF + poly for interpolation and derivative approximation. J. Sci. Comput. 81(1), 486\u2013512 (2019)","journal-title":"J. Sci. Comput."},{"key":"1065_CR4","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1016\/j.jcp.2018.12.013","volume":"380","author":"V Bayona","year":"2019","unstructured":"Bayona, V., Flyer, N., Fornberg, B.: On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries. J. Comput. Phys. 380, 378\u2013399 (2019)","journal-title":"J. Comput. Phys."},{"key":"1065_CR5","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1016\/j.jcp.2016.12.008","volume":"332","author":"V Bayona","year":"2017","unstructured":"Bayona, V., Flyer, N., Fornberg, B., Barnett, G.: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J. Comput. Phys. 332, 257\u2013273 (2017)","journal-title":"J. Comput. Phys."},{"key":"1065_CR6","unstructured":"Candes, E., Romberg, J.: L1-magic: Recovery of sparse signals via convex programming, vol. 4. \nwww.acm.caltech.edu\/l1magic\/downloads\/l1magic.pdf\n\n (2005)"},{"key":"1065_CR7","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.jcp.2015.01.018","volume":"285","author":"A Demirel","year":"2015","unstructured":"Demirel, A., Niegemann, J., Busch, K., Hochbruck, M.: Efficient multiple time-stepping algorithms of higher order. J. Comput. Phys. 285, 133\u2013148 (2015)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1065_CR8","first-page":"190","volume":"56","author":"S Descombes","year":"2013","unstructured":"Descombes, S., Lanteri, S., Moya, L.: Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell\u2019s equations. J. Comput. Phys. 56(1), 190\u2013218 (2013)","journal-title":"J. Comput. Phys."},{"key":"1065_CR9","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1016\/j.cma.2015.03.027","volume":"291","author":"J Diaz","year":"2015","unstructured":"Diaz, J., Grote, M.: Multi-level explicit local time-stepping methods for second-order wave equations. Comput. Methods Appl. Mech. Eng. 291, 240\u2013265 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"6","key":"1065_CR10","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1016\/j.camwa.2006.06.005","volume":"53","author":"TA Driscoll","year":"2007","unstructured":"Driscoll, T.A., Heryudono, A.R.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53(6), 927\u2013939 (2007)","journal-title":"Comput. Math. Appl."},{"key":"1065_CR11","doi-asserted-by":"publisher","DOI":"10.1142\/6437","volume-title":"Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences","author":"G Fasshauer","year":"2007","unstructured":"Fasshauer, G.: Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishers, Singapore (2007)"},{"key":"1065_CR12","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/j.jcp.2016.02.078","volume":"316","author":"N Flyer","year":"2016","unstructured":"Flyer, N., Barnett, G., Wicker, L.: Enhancing finite differences with radial basis functions: experiments on the Navier\u2013Stokes equations. J. Comput. Phys. 316, 39\u201362 (2016)","journal-title":"J. Comput. Phys."},{"key":"1065_CR13","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.jcp.2016.05.026","volume":"321","author":"N Flyer","year":"2016","unstructured":"Flyer, N., Fornberg, B., Barnett, G., Bayona, V.: On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J. Comput. Phys. 321, 21\u201338 (2016)","journal-title":"J. Comput. Phys."},{"key":"1065_CR14","doi-asserted-by":"publisher","first-page":"4078","DOI":"10.1016\/j.jcp.2012.01.028","volume":"231","author":"N Flyer","year":"2012","unstructured":"Flyer, N., Lehto, E., Blaise, S., Wright, G., St-Cyr, A.: A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. J. Comput. Phys. 231, 4078\u20134095 (2012)","journal-title":"J. Comput. Phys."},{"key":"1065_CR15","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.camwa.2015.01.009","volume":"69","author":"B Fornberg","year":"2015","unstructured":"Fornberg, B., Flyer, N.: Fast generation of 2-D node distributions for mesh-free PDE discretizations. Comput. Math. Appl. 69, 531\u2013544 (2015)","journal-title":"Comput. Math. Appl."},{"key":"1065_CR16","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974041","volume-title":"A Primer on Radial Basis Functions with Applications to the Geosciences","author":"B Fornberg","year":"2015","unstructured":"Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM, Philadelphia (2015)"},{"key":"1065_CR17","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1017\/S0962492914000130","volume":"24","author":"B Fornberg","year":"2015","unstructured":"Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numerica 24, 215\u2013258 (2015)","journal-title":"Acta Numerica"},{"key":"1065_CR18","doi-asserted-by":"publisher","first-page":"2270","DOI":"10.1016\/j.jcp.2010.12.014","volume":"230","author":"B Fornberg","year":"2011","unstructured":"Fornberg, B., Lehto, E.: Stabilization of RBF-generated finite difference methods for convective PDEs. J. Comput. Phys. 230, 2270\u20132285 (2011)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"1065_CR19","doi-asserted-by":"publisher","first-page":"B1043","DOI":"10.1137\/16M1101374","volume":"39","author":"J Gopalakrishnan","year":"2017","unstructured":"Gopalakrishnan, J., Sch\u00f6berl, J., Wintersteiger, C.: Mapped tent pitching schemes for hyperbolic systems. SIAM J. Sci. Comput. 39(6), B1043\u2013B1063 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1065_CR20","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/j.enganabound.2016.03.009","volume":"67","author":"M Hamaidi","year":"2016","unstructured":"Hamaidi, M., Naji, A., Charafi, A.: Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations. Eng. Anal. Bound. Elem. 67, 152\u2013163 (2016)","journal-title":"Eng. Anal. Bound. Elem."},{"issue":"8","key":"1065_CR21","doi-asserted-by":"publisher","first-page":"3442","DOI":"10.1016\/j.apm.2008.11.020","volume":"33","author":"Sirajul Haq","year":"2009","unstructured":"Haq, S., Siraj-Ul-Islam, Uddin, M.: A mesh-free method for the numerical solution of the KdV-Burgers equation. Appl. Mathe. Model. 33, 3442\u20133449 (2008)","journal-title":"Applied Mathematical Modelling"},{"issue":"12","key":"1065_CR22","doi-asserted-by":"publisher","first-page":"1611","DOI":"10.1016\/j.envsoft.2011.07.010","volume":"26","author":"Z Li","year":"2011","unstructured":"Li, Z., Mao, X.Z.: Global multiquadric collocation method for groundwater contaminant source identification. Environ. Model. Softw. 26(12), 1611\u20131621 (2011)","journal-title":"Environ. Model. Softw."},{"key":"1065_CR23","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.advwatres.2015.11.019","volume":"88","author":"Z Li","year":"2016","unstructured":"Li, Z., Mao, X.Z., Li, T.S., Zhang, S.: Estimation of river pollution source using the space-time radial basis collocation method. Adv. Water Resour. 88, 68\u201379 (2016)","journal-title":"Adv. Water Resour."},{"issue":"3","key":"1065_CR24","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1002\/nme.2638","volume":"80","author":"H Netuzhylov","year":"2009","unstructured":"Netuzhylov, H., Zilian, A.: Space-time meshfree collocation method: methodology and application to initial-boundary value problems. Int. J. Numer. Meth. Eng. 80(3), 355\u2013380 (2009)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"1065_CR25","first-page":"99","volume":"25","author":"Y Shan","year":"2008","unstructured":"Shan, Y., Shu, C., Lu, Z.: Application of local MQ-DQ method to solve 3D incompressible viscous flows with curved boundary. Comput. Model. Eng. Sci. 25, 99\u2013113 (2008)","journal-title":"Comput. Model. Eng. Sci."},{"issue":"10","key":"1065_CR26","doi-asserted-by":"publisher","first-page":"212","DOI":"10.3390\/math6100212","volume":"6","author":"M Uddin","year":"2018","unstructured":"Uddin, M., Ali, H.: The space-time kernel-based numerical method for Burgers\u2019 equations. Mathematics 6(10), 212 (2018)","journal-title":"Mathematics"},{"key":"1065_CR27","volume-title":"Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics","author":"H Wendland","year":"2005","unstructured":"Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)"},{"key":"1065_CR28","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/j.jcp.2005.05.030","volume":"212","author":"G Wright","year":"2006","unstructured":"Wright, G., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212, 99\u2013123 (2006)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01065-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-01065-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01065-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T23:24:56Z","timestamp":1602717896000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-01065-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,16]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["1065"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-01065-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,16]]},"assertion":[{"value":"11 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}